arXiv:1105.5362v1 [nlin.CD] 26 May 2011
Resonance-assisted tunneling in
mixed regular-chaotic systems
Peter Schlagheck
1, Amaury Mouchet
2, and Denis Ullmo
31 D´epartement de Physique, Universit´e de Li`ege, 4000 Li`ege, Belgium
2 Laboratoire de Math´ematiques et de Physique Th´eorique, Universit´e Fran¸cois Rabelais de
Tours — cnrs (umr 6083), F´ed´eration Denis Poisson, Parc de Grandmont, 37200 Tours, France
3 LPTMS UMR 8626, Univ. Paris-Sud, CNRS, 91405 Orsay Cedex, France
8.1
Introduction
8.1.1 Tunneling in integrable systems
Since the early days of quantum mechanics, tunneling has been recognized as one of the hall-marks of the wave character of microscopic physics. The possibility of a quantum particle to penetrate an energetic barrier represents certainly one of the most spectacular implications of quantum theory and has lead to various applications in nuclear, atomic and molecular physics as well as in mesoscopic science. Typical scenarios in which tunneling manifests are the escape of a quantum particle from a quasi-bounded region, the transition between two or more symmetry-related, but classically disconnected wells (which we shall focus on in the following), as well as scattering or transport through potential barriers. The spectrum of scenarios becomes even richer when the concept of tunneling is generalized to any kind of classically forbidden transitions in phase space, i.e. to transitions that are not necessarily inhibited by static potential barriers but by some other constraints of the underlying clas-sical dynamics (such as integrals of motion). Such “dynamical tunneling” processes arise frequently in molecular systems [1] and were realized with cold atoms propagating in period-ically modulated optical lattices [2, 3, 4]. Moreover, the electromagnetic analog of dynamical
tunneling was also obtained with microwaves in billiards [5].
Despite its genuinely quantal nature, tunneling is strongly influenced by the structure of the underlying classical phase space (see Ref. [6] for a review). This is best illustrated within the textbook example of a one-dimensional symmetric double-well potential. In this simple case, the eigenvalue problem can be straightforwardly solved with the standard Jeffreys-Wentzel-Kramers-Brillouin (JWKB) ansatz [7]. The eigenstates of this system are, below the barrier height, obtained by the symmetric and antisymmetric linear combination of the local “quasi-modes” (i.e., of the wave functions that are semiclassically constructed on the quantized orbits within each well, without taking into account the classically forbidden coupling between the wells), and the splitting of their energies is given by an expression of the form ∆E = ~Ω π exp −~1 Z p2m(V (x) − E)dx . (8.1)
Here E is the mean energy of the doublet, V (x) represents the double well potential, m is the mass of the particle, Ω denotes the oscillation frequency within each well, and the integral in the exponent is performed over the whole classically forbidden domain, i.e. between the inner turning points of the orbits in the two wells. Preparing the initial state as one of the quasi-modes (i.e., as the even or odd superposition of the symmetric and the antisymmetric eigenstate), the system will undergo Rabi oscillations between the wells with the frequency ∆E/~. The “tunneling rate” of this system is therefore given by the splitting (8.1). Keeping all classical parameters fixed, it decreases exponentially with 1/~, and, in that sense, one can say that tunneling “vanishes” in the classical limit.
8.1.2 Chaos-assisted tunneling
The approach presented in the previous section can be generalized to multidimensional, even non-separable systems, as long as their classical dynamics is still integrable [8]. It breaks down, however, as soon as a non-integrable perturbation is added to the system, e.g. if the one-dimensional double-well potential is exposed to a driving that is periodic in time (with period τ , say). In that case, the classical phase space of the system generally becomes a mixture of both regular and chaotic structures.
As visualized by the stroboscopic Poincar´e section — which is obtained by retaining the phase space coordinates at every integer multiple of the driving period τ — the phase space typically displays two prominent regions of regular motion, corresponding to the weakly per-turbed dynamics within the two wells, and a small (or, for stronger perturbations, large) layer of chaotic dynamics that separates the two regular islands from each other. Numerical calculations of model systems in the early nineties [9, 10] have shown that the tunnel split-tings in such mixed systems generally become strongly enhanced compared to the integrable limit. Moreover, they do no longer follow a smooth exponential scaling with 1/~ as expressed by Eq. (8.1), but display huge, quasi-erratic fluctuations when ~ or any other parameter of the system vary [9, 10].
These phenomena are traced back to the specific role that chaotic states play in such systems [11, 12, 13, 14]. In contrast to the integrable case, the tunnel doublets of the localized quasi-modes are, in a mixed regular-chaotic system, no longer isolated in the spectrum, but resonantly interact with states that are associated with the chaotic part of phase space. Due
to their delocalized nature in phase space1, such chaotic states typically exhibit a significant
overlap with the boundary regions of both regular wells. Therefore, they may provide an efficient coupling mechanism between the quasi-modes – which becomes particularly effective whenever one of the chaotic levels is shifted exactly on resonance with the tunnel doublet. As illustrated in Fig. 8.1, this coupling mechanism generally enhances the tunneling rate, but may also lead to a complete suppression thereof, arising at specific values of ~ or other parameters [16].
We point out that this type of resonant tunneling does not necessarily require the presence of classical chaos and may appear also in integrable systems, for instance in a one-dimensional symmetric triple-well potential. This is illustrated in Fig. 8.2, which shows the scaling of level splittings associated with the two lateral wells as a function of 1/~ for such a triple-well potential. On top of an exponential decrease according to Eq. (8.1), the splittings display strong spikes occuring whenever the energy of a state localized in the central well becomes quasi-degenerate with the energies of the states in the two lateral wells.
a) b) external parameter ∆ǫ (log scale) ǫ + − + − + +
Figure 8.1: The two elementary scenarios of huge enhancement (case b) or cancellation (case a) of the tunneling splitting between symmetric (+) and anti-symmetric (−) states can be easily understood from the resonant crossing of a third level (here a symmetric one) and the corresponding level repulsion between states of the same symmetry class. The external parameter that triggers the fluctuation of tunneling can be of quantum origin (effective ~) or classical.
For mixed regular-chaotic systems, the validity of this “chaos-assisted” tunneling picture was essentially confirmed by successfully modeling the chaotic part of the quantum dynamics with a random matrix from the Gaussian orthogonal ensemble (GOE) [11, 12, 17]. Using the fact that the coupling coefficients between the regular states and the chaotic domain are small, this random matrix ansatz yields a truncated Cauchy distribution for the probability density to obtain a level splitting of the size ∆E. Such a distribution is indeed encountered in the exact quantum splittings, which was demonstrated for the two-dimensional quartic oscillator [17] as well as, later on, for the driven pendulum Hamiltonian that describes
1
2
3
4
5
6
7
-30
-25
-20
-15
-10
-5
0
V (x) x1/~
∆Eln ∆E
Figure 8.2: Resonant tunneling at work for a one dimensional triple well potential (here
V (x) = (x2 − a2)2(x2 − b2) with a = 1.75 and b = .5). When 1/~ is increased, tunneling
between the two lateral wells is enhanced by several orders of magnitude each time a level in the central well crosses down the doublet.
the tunneling process of cold atoms in periodically modulated optical lattices [4, 18]. A quantitative prediction of the average tunneling rate, however, was not possible in the above-mentioned theoretical works. As we shall argue later on, this average tunneling rate is directly connected to the coupling matrix element between the regular and the chaotic states, and the strength of this matrix element was unknown and introduced in an ad-hoc way.
A first step towards this latter problem was undertaken by Podolskiy and Narimanov [19] who proposed an explicit semiclassical expression for the mean tunneling rate in a mixed system by assuming a perfectly clean, harmonic-oscillator like dynamics within the regular island and a structureless chaotic sea outside the outermost invariant torus of the island. This expression turned out to be successful for the reproduction of the level splittings between near-degenerate optical modes that are associated with a pair of symmetric regular islands in a non-integrable micro-cavity [19] (see also Ref. [20]). The application to dynamical tunneling in periodically modulated optical lattices [19], for which splittings between the left- and the right-moving stable eigenmodes were calculated in Ref. [4], seems convincing for low and moderate values of 1/~, but reveals deviations deeper in the semiclassical regime where plateau structures arise in the tunneling rates. Further, and more severe, deviations were encountered in the application of this approach to tunneling processes in other model systems [21].
B¨acker, Ketzmerick, L¨ock, and coworkers [22, 23] recently undertook the effort to de-rive more rigorously the regular-to-chaotic coupling rate governing chaos-assisted tunneling. Their approach is based on the construction of an integrable approximation for the noninte-grable system, designed to accurately describe the motion within the regular islands under consideration. The coupling rate to the chaotic domain is then determined through the com-putation of matrix elements of the Hamiltonian of the system within the eigenbasis of this integrable approximation [22]. This results in a smooth exponential-like decay of the aver-age tunneling rate with 1/~, which was indeed found to be in very good agreement with the exact tunneling rates for quantum maps and billiards [22, 23]. Those systems, however, were designed such as to yield a “clean” mixed regular-chaotic phase space, containing a regular island and a chaotic region which both do not exhibit appreciable substructures [22, 23].
8.1.3 The role of nonlinear resonances
In more generic systems, such as the quantum kicked rotor or the driven pendulum [4], however, even the “average” tunneling rates do not exhibit a smooth monotonous behaviour with 1/~, but display peaks and plateau structures that cannot be accounted for by the above approaches. To understand the origin of such plateaus, it is instructive to step back to the conceptually simpler case of nearly integrable dynamics, where the perturbation from the integrable Hamiltonian is sufficiently small such that macroscopically large chaotic layers are not yet developed in the Poincar´e surface of section. In such systems, the main classical phase space features due to the perturbation consist in chain-like substructures that surround stable periodic orbits or equilibrium points of the classical motion. Those substructures come from nonlinear resonances between the internal degrees of freedom of the system or, for driven systems, between the external driving and the unperturbed oscillations around the central orbit. In a similar way as for the quantum pendulum Hamiltonian, such resonances induce additional tunneling paths in the phase space, which lead to couplings between states that are located near the same stable orbit [24, 25].
The relevance of this effect for the near-integrable tunneling process between two symmetry-related wells was first pointed out by Bonci et al. [26] who argued that such resonances may lead to a strong enhancement of the tunneling rate, due to couplings between lowly and highly excited states within the well which are permitted by near-degeneracies in the spec-trum In Refs. [27, 28], a quantitative semiclassical theory of near-integrable tunneling was formulated on the basis of this principal mechanism. This theory allows one to reproduce the exact quantum splittings from purely classical quantities and takes into account high-order effects such as the coupling via a sequence of different resonance chains [27, 28]. More re-cent studies by Keshavamurthy on classically forbidden coupling processes in model systems that mimic the dynamics of simple molecules confirm that the resonance-assisted tunneling scenario prevails not only in one-dimensional systems that are subject to a periodic driving (such as the kicked Harper model studied in Ref. [27, 28]), but also in autonomous systems with two and even three degrees of freedom [29, 30].
In Refs. [31, 32, 33, 34] resonance-assisted couplings were incorporated in an approximate manner into the framework of chaos-assisted tunneling in order to provide a quantitative the-ory for the regular-to-chaotic coupling rate. In this context, it is assumed that the dominant coupling between regular states within and chaotic states outside the island is provided by the presence of a nonlinear resonance within the island. A straightforward implementation of this idea yields good agreement with the exact tunneling rates as far as their average decay with 1/~ in the deep semiclassical limit is concerned. Moreover, individual plateaus
and peak structures could be traced back to the influence of specific nonlinear resonances, not only for double-well-type tunneling in closed or periodic systems [31, 32, 33], but also for tunneling-induced decay in open systems [34]. However, the predictive power of this method was still rather limited, insofar as individual tunneling rates at given system pa-rameters could be over- or underestimated by many orders of magnitude. In particular, resonance-assisted tunneling seemed inapplicable in the “quantum” limit of large ~, where direct regular-to chaotic tunneling proved successful [22, 23].
A major advance in this context was achieved by improving the semiclassical evaluation of resonance-induced coupling processes in mixed systems, and by combining it with “direct” regular-to-chaotic tunneling [35]. This combination resulted, for the first time, in a semi-classical prediction of tunneling rates in generic mixed regular-chaotic systems that can be compared with the exact quantum rates on the level of individual peak structures [35]. This confirms the expectation that nonlinear resonances do indeed form the “backbone” behind non-monotonous substructures in tunneling rates. It furthermore suggests that those rates could, also in systems with more degrees of freedom, possibly be estimated in a quantita-tively satisfactory manner via simple classical computations, based on the most prominent nonlinear resonances that are manifested within the regular island.
It is in the spirit of this latter expectation that this contribution has been written. Our aim is not to formulate a formal semiclassical theory of tunneling in mixed systems, which still represents an open problem that would rather have to be solved on the basis of complex classical orbits [36, 37, 38]. Instead, we want to provide a simple, easy-to-implement, yet effective prescription how to compute the rates and time scales associated with tunneling processes solely on the basis of the classical dynamics of the system, without performing any diagonalization (not even any application) of the quantum Hamiltonian or of the time evolution operator. This prescription is based on chaos- and resonance-assisted tunneling in its improved form [35]. The main part of this contribution is therefore devoted to a detailed description of resonance-assisted tunneling and its combination with chaos-assisted tunneling in the sections 8.2 and 8.3, respectively. We present in section 8.4 the application of this method to tunneling processes in the quantum kicked rotor, and discuss possible limitations and future prospects in the conclusion in section 8.5.
8.2
Theory of resonance-assisted tunneling
8.2.1 Secular perturbation theory
For our study, we restrict ourselves to systems with one degree of freedom that evolve under a periodically time-dependent Hamiltonian H(p, q, t) = H(p, q, t + τ ). We suppose that, for a suitable choice of parameters, the classical phase space of H is mixed regular-chaotic and exhibits two symmetry-related regular islands that are embedded within the chaotic sea. This phase space structure is most conveniently visualized by a stroboscopic Poincar´e section, where p and q are plotted at the times t = nτ (n ∈ Z). Such a Poincar´e section typically reveals the presence of chain-like substructures within the regular islands, which arise due to nonlinear resonances between the external driving and the internal oscillation around the island’s center. Before considering the general situation for which many resonances may come into play in the tunneling process, we start with the simpler case where the two islands exhibit a prominent r:s resonance, i.e., a nonlinear resonance where s internal oscillation
periods match r driving periods and r sub-islands are visible in the stroboscopic section. The classical motion in the vicinity of the r:s resonance is approximately integrated by secular perturbation theory [39] (see also Ref. [28]). For this purpose, we formally introduce
a time-independent Hamiltonian H0(p, q) that approximately reproduces the regular motion
in the islands and preserves the discrete symmetry of H. In some circumstances, as for
instance if H is in the nearly integrable regime, H0(p, q) can be explicitly computed within
some approximation scheme (using for instance the Lie transformation method [39]). We
stress though that this will not always be necessary. Assuming the existence of such a H0, the
phase space generated by this integrable Hamiltonian consequently exhibits two symmetric wells that are separated by a dynamical barrier and “embed” the two islands of H. In terms of the action-angle variables (I, θ) describing the dynamics within each of the wells, the total Hamiltonian can be written as
H(I, θ, t) = H0(I) + V (I, θ, t) (8.2)
where V would represent a weak perturbation in the center of the island 2
The nonlinear r:s resonance occurs at the action variable Ir:s that satisfies the condition
rΩr:s= sω (8.3) with ω = 2π/τ and Ωr:s≡ dH0 dI I=Ir:s . (8.4)
We now perform a canonical transformation to the frame that corotates with this resonance. This is done by leaving I invariant and modifying θ according to
θ 7→ ϑ = θ − Ωr:st . (8.5)
This time-dependent shift is accompanied by the transformation H 7→ H = H − Ωr:sI in
order to ensure that the new corotating angle variable ϑ is conjugate to I. The motion of I and ϑ is therefore described by the new Hamiltonian
H(I, ϑ, t) = H0(I) + V(I, ϑ, t) (8.6)
with
H0(I) = H0(I) − Ωr:sI , (8.7)
V(I, ϑ, t) = V (I, ϑ + Ωr:st, t) . (8.8)
The expansion of H0 in powers of I − Ir:s yields
H0(I) ≃ H(0)0 +(I − I
r:s)2
2mr:s + O(I − Ir:s
)3
(8.9)
with a constant H(0)0 ≡ H0(Ir:s) − Ωr:sIr:s and a quadratic term that is characterized by
the effective “mass” parameter mr:s ≡ [d2H0/dI2(Ir:s)]−1. Hence, dH0/dI is comparatively
small for I ≃ Ir:s, which implies that the co-rotating angle ϑ varies slowly in time near
2
In order not to overload the notation, we use the same symbol H for the Hamiltonian in the original phase-space variables (p, q) and in the action-angle variables (I, θ).
the resonance. This justifies the application of adiabatic perturbation theory [39], which effectively amounts, in first order, to replacing V(I, ϑ, t) by its time average over r periods
of the driving (using the fact that V is periodic in t with the period rτ) 3. We therefore
obtain, after this transformation, the time-independent Hamiltonian
H(I, ϑ) = H0(I) + V(I, ϑ) (8.10)
with
V(I, ϑ) ≡ rτ1
Z rτ
0 V(I, ϑ, t)dt .
(8.11)
By expanding V (I, θ, t) in a Fourier series in both θ and t, i.e.
V (I, θ, t) =
∞
X
l,m=−∞
Vl,m(I)eilθeimωt (8.12)
with Vl,m(I) = [V−l,−m(I)]∗, one can straightforwardly derive
V(I, ϑ) = V0,0(I) + ∞ X k=1 2Vk(I) cos(krϑ + φk) (8.13) defining
Vk(I)eiφk ≡ Vrk,−sk(I) , (8.14)
i.e., the resulting time-independent perturbation term is (2π/r)-periodic in ϑ.
For the sake of clarity, we start discussing the resulting effective Hamiltonian neglecting the action dependence of the Fourier coefficients of V(I, ϑ). We stress that this dependence can be implemented in a relatively straightforward way using Birkhoff-Gustavson normal-form coordinates (cf section 8.2.3 below); it is actually important to obtain a good
quan-titative accuracy. For now, however, we replace Vk(I) by Vk ≡ Vk(I = Ir:s) in Eq. (8.13).
Neglecting furthermore the term V0,0(I), we obtain the effective integrable Hamiltonian
Hres(I, ϑ) = H0(I) − Ωr:sI +
∞
X
k=1
2Vkcos(krϑ + φk) (8.15)
for the description of the classical dynamics in the vicinity of the resonance. We shall see in
section 8.2.2 that the parameters of Hres relevant to the tunneling process can be extracted
directly from the classical dynamics of H(t), which is making Eq. (8.15) particularly valuable.
8.2.2 The pendulum approximation
The quantum implications due to the presence of this nonlinear resonance can be
straight-forwardly inferred from the direct semiclassical quantization of Hres, given by
ˆ Hres= H0( ˆI) − Ωr:sI +ˆ ∞ X k=1 2Vkcos(kr ˆϑ + φk) . (8.16) 3
This step involves, strictly speaking, another time-dependent canonical transformation (I, ϑ) 7→ (eI, eϑ) which slightly modifies I and ϑ (see also Ref. [28]).
Here we introduce the action operator ˆI ≡ −i~∂/∂ϑ and assume anti-periodic boundary conditions in ϑ in order to properly account for the Maslov index in the original phase space [24]. In accordance with our assumption that the effect of the resonance is rather weak,
we can now apply quantum perturbation theory to the Hamiltonian (8.16), treating the ˆ
I-dependent “kinetic” terms as unperturbed part and the ˆϑ-dependent series as perturbation.
The unperturbed eigenstates are then given by the (anti-periodic) eigenfunctions hϑ|ni =
(2π)−1/2exp[i(n + 1/2)ϑ] (n ≥ 0) of the action operator ˆI with the eigenvalues
In = ~(n + 1/2) . (8.17)
As is straightforwardly evaluated, the presence of the perturbation induces couplings between the states |ni and |n + kri with the matrix elements
hn + kr| ˆHres|ni = Vkeiφk (8.18)
for strictly positive integer k. As a consequence, the “true” eigenstates |ψni of ˆHres contain
admixtures from unperturbed modes |n′i that satisfy the selection rule |n′ − n| = kr with
integer k. They are approximated by the expression
|ψni = |ni + X k hn + kr| ˆHres|ni En− En+kr + ks~ω|n + kri + +X k,k′ hn + kr| ˆHres|n + k′ri En− En+kr + ks~ω hn + k′r| ˆH res|ni En− En+k′r+ k′s~ω|n + kri + . . . (8.19)
where En ≡ H0(In) denote the unperturbed eigenenergies of H0 and the resonance condition
(8.3) is used. The summations in Eq. (8.19) are generally finite due to the finiteness of the phase space area covered by the regular region.
Within the quadratic approximation of H0(I) around Ir:s, we obtain from Eqs. (8.7) and
(8.9)
En≃ H0(Ir:s) + Ωr:s(In− Ir:s) +
1
2mr:s
(In− Ir:s)2. (8.20)
This results in the energy differences
En− En+kr + ks~ω ≃
1
2mr:s
(In− In+kr)(In+ In+kr− 2Ir:s) . (8.21)
From this expression, we see that the admixture between |ni and |n′i becomes particularly
strong if the r:s resonance is symmetrically located between the two tori that are associated
with the actions In and In′ — i.e., if In+ In′ ≃ 2Ir:s. The presence of a significant nonlinear
resonance within a region of regular motion provides therefore an efficient mechanism to couple the local “ground state” — i.e, the state that is semiclassically localized in the center
of that region (with action variable I0 < Ir:s) — to a highly excited state (with action
variable Ikr > Ir:s).
It is instructive to realize that the Fourier coefficients Vk of the perturbation operator
decrease rather rapidly with increasing k. Indeed, one can derive under quite general cir-cumstances the asymptotic scaling law
for large k, which is based on the presence of singularities of the complexified tori of the
integrable approximation H0(I) (see Eq. (66) in Ref. [28]). Here tim(I) denotes the imaginary
time that elapses from the (real) torus with action I to the nearest singularity in complex
phase space, γ corresponds to the degree of the singularity, and V0contains information about
the corresponding residue near the singularity as well as the strength of the perturbation. The expression (8.22) is of little practical relevance as far as the concrete determination of
the coefficients Vk is concerned. It permits, however, to estimate the relative importance
of different perturbative pathways connecting the states |ni and |n + kri in Eq. (8.19).
Comparing e.g. the amplitude A2 associated with a single step from |ni to |n + 2ri via V2
and the amplitude A1 associated with two steps from |ni to |n + 2ri via V1, we obtain from
Eqs. (8.21) and (8.22) the ratio
A2/A1 ≃
2γ−1r2−γ~2
mr:sV0
ei(φ2−2φ1)
(8.23)
under the assumption that the resonance is symmetrically located in between the
correspond-ing two tori (in which case we would have In+r ≃ Ir:s). Since V0 can be assumed to be finite
in mixed regular-chaotic systems, we infer that the second-order process via the stronger
coefficient V1 will more dominantly contribute to the coupling between |ni and |n + 2ri in
the semiclassical limit ~ → 0.
A similar result is obtained from a comparison of the one-step process via Vk with the
k-step process via V1, where we again find that the latter more dominantly contributes to
the coupling between |ni and |n + kri in the limit ~ → 0. We therefore conclude that in mixed regular-chaotic systems the semiclassical tunneling process is adequately described by
the lowest non-vanishing term of the sum over the Vkcontributions, which in general is given
by V1cos(rϑ + φ1)4. Neglecting all higher Fourier components Vk with k > 1 and making the
quadratic approximation of H0 around I = Ir:s, we finally obtain an effective pendulum-like
Hamiltonian Hres(I, ϑ) ≃ (I − Ir:s )2 2mr:s + 2Vr:scos(rϑ + φ1) (8.24) with Vr:s≡ V1 [31].
This simple form of the effective Hamiltonian allows us to determine the parameters Ir:s,
mr:s and Vr:s from the Poincar´e map of the classical dynamics, without explicitly using the
transformation to the action-angle variables of H0. To this end, we numerically calculate
the monodromy matrix Mr:s≡ ∂(pf, qf)/∂(pi, qi) of a stable periodic point of the resonance
(which involves r iterations of the stroboscopic map) as well as the phase space areas S+
r:s
and S−
r:s that are enclosed by the outer and inner separatrices of the resonance, respectively
(see also Fig. 8.3). Using the fact that the trace of Mr:s as well as the phase space areas Sr:s±
remain invariant under the canonical transformation to (I, ϑ), we infer
Ir:s = 1 4π(S + r:s+ Sr:s−) , (8.25) p2mr:sVr:s = 1 16(S + r:s− Sr:s− ) , (8.26) r 2Vr:s mr:s = 1 r2τ arccos(tr Mr:s/2) (8.27) 4
Exceptions from this general rule typically arise in the presence of discrete symmetries that, e.g., forbid the formation of resonance chains with an odd number of sub-islands and therefore lead to V1= 0 for an r:s resonance with an odd r.
from the integration of the dynamics generated by Hres [40]. Eqs. (8.25)-(8.27) make it
possible to derive the final expressions for the tunneling rates directly from the dynamics of
H(t), without explicitly having to construct the integrable approximation H0 and making
the Fourier analysis of V (p, q, t) = H(p, q, t)−H0(p, q). As this construction of the integrable
approximation may turn out to be highly non-trivial in the mixed regime, avoiding this step is actually an essential ingredient to make the approach we are following practical. We note
though that improving the quadratic approximation (8.24) for H0 is sometimes necessary,
but this does not present any fundamental difficulty.
Figure 8.3: Classical phase space of the kicked rotor Hamiltonian at K = 3.5 showing a regular island with an embedded 6:2 resonance. The phase space is plotted in the original (p, q) coordinates (upper left panel), in approximate normal-form coordinates (P, Q) defined by Eqs. (8.28) and (8.29) (upper right panel), and in approximate action-angle variables (I, ϑ) (lower panel). The thick solid and dashed lines represent the “outer” and “inner” separatrix of the resonance, respectively.
8.2.3 Action dependence of the coupling coefficients
Up to now, and in our previous publications [27, 28, 31, 32, 33], we completely neglected the
action dependence of the coupling coefficients Vk(I). This approximation should be justified
in the semiclassical limit of extremely small ~, where resonance-assisted tunneling generally involves multiple coupling processes [28] and transitions across individual resonance chains are therefore expected to take place in their immediate vicinity in action space. For finite
~, however, the replacement Vk(I) 7→ Vk(Ir:s), permitting the direct quantization in
action-angle space, is, in general, not sufficient to obtain an accurate reproduction of the quantum tunneling rates. We show now how this can be improved.
To this end, we make the general assumption that the classical Hamiltonian H(p, q, t) of our system is analytic in p and q in the vicinity of the regular islands under consideration. It is then possible to define an analytical canonical transformation from (p, q) to Birkhoff-Gustavson normal-form coordinates (P, Q) [41, 42] that satisfy
P = −√2I sin θ , (8.28)
Q = √2I cos θ (8.29)
and that can be represented in power series in p and q. The “unperturbed” integrable
Hamiltonian H0 therefore depends only on I = (P2+ Q2)/2.
Writing
e±ilθ = Q ∓ iP√
2I
l
(8.30)
for positive l, we obtain, from Eq. (8.12), the series
V (I, θ, t) = ∞ X m=−∞ ( V0,m(I) + ∞ X l=1 1 √ 2Il Vl,m(I)(Q − iP )l+ V−l,m(I)(Q + iP )l ) eimωt (8.31) for the perturbation. Using the fact that V (I, θ, t) is analytic in P and Q, we infer that
Vl,m(I) must scale at least as Il/2. By virtue of (8.14), this implies the scaling Vk(I) ∝ Irk/2
for the Fourier coefficients of the time-independent perturbation term that is associated with
the r:s resonance. Making the ansatz Vk(I) ≡ Irk/2˜vk (and neglecting the residual action
dependence of ˜vk), we rewrite Eq. (8.13) as5
V(I, ϑ) = V0,0(I) + ∞ X k=1 ˜ vk 2kr/2(Q − iP ) kreiφk + (Q + iP )kre−iφk . (8.32)
Each term in the sum is given by the well-known Birkhoff normal form generically describing a r ≥ 3 resonance when the bifurcation of the stable periodic orbit is controlled by one single parameter (see, e.g., Eq. (4.70) in Chapter 4 of Ref. [43] or Eqs. (3.3.17) and (3.3.18) in
Ref. [44] for another simple derivation of the action dependence of Vk(I)). The term V0,0(I)
is neglected in the following as it does not lead to any coupling between different unperturbed eigenstates in the quantum system.
5
One comment is in order here. The small parameter in the adiabatic approximation (8.10)
is the difference (I−Ir:s), while in the derivation of Eq. (8.32) we have neglected higher powers
of I and thus assumed the action I itself to be small (strictly speaking we should work near one bifurcation only). We thus mix a development near the resonant torus with one near the center of the island. This may eventually become problematic if (i) the resonance chain is located far away from the center of the island, in which case the associated coupling strength may contain a nonnegligible relative error when being computed via the assumption
Vk(I) ≡ Irk/2˜vk with constant ˜vk, and if (ii) that coupling strength happens to appear rather
often in the main perturbative chain that connects the quasimodes of the island to the chaotic sea, which generally would be the case for low-order resonances with relatively small
r.6 Otherwise, we expect that this inconsistency in the definition of the regimes of validity
of our perturbative approach does not lead to a significant impact on the numerical values of the semiclassical tunneling rates, which indeed seems to be confirmed by numerical evidence to be discussed below.
This being said, the quantization of the resulting classical Hamiltonian can now be carried out in terms of the “harmonic oscillator” variables P and Q and amounts to introducing the
standard ladder operators ˆa and ˆa† according to
ˆa = √1
2~( ˆQ + i ˆP ) , (8.33)
ˆa† = √1
2~( ˆQ − i ˆP ) . (8.34)
This yields the quantum Hamiltonian
ˆ Hres = H0( ˆI) − Ωr:sI +ˆ ∞ X k=1 ˜
vk~kr/2ˆakre−iφk+ (ˆa†)kreiφk
(8.35)
with ˆI ≡ ~(ˆa†a+1/2). As for Eq. (8.16), perturbative couplings are introduced only betweenˆ
unperturbed eigenstates |ni and |n′i that exhibit the selection rule |n′− n| = kr with integer
k. The associated coupling matrix elements are, however, different from Eq. (8.18) and read
hn + kr| ˆHres|ni = ˜vk √ ~kreiφk r (n + kr)! n! = Vk(Ir:s)eiφk ~ Ir:s kr/2r (n + kr)! n! (8.36)
for strictly positive k. Close the resonance, i.e. more formally taking the semiclassical
limit n → ∞ keeping k and δ = (Ir:s/~−n) fixed, and making use of the Stirling formula
n! ≃ √2πn(n/e)n, Eq. (8.36) reduces to V
k(Ir:s)eiφk. The difference becomes, on the other
hand, particularly pronounced if the r:s resonance is, in phase space, rather asymmetrically located in between the invariant tori that correspond to the states |ni and |n + kri — i.e.,
if Ir:s is rather close to In or to In+kr. In that case, Eq. (8.18) may, respectively, strongly
over- or underestimate the coupling strength between these states.
6
Corrections to the form (8.32) should also arise in the presence of prominent secondary resonances, which occur when primary resonances start to overlap and create chains of sub-islands nested inside the primary island chains.
8.2.4 Multi-resonance processes
Up to this point, we considered the couplings between quasi-modes generated by a given resonance. In general, however, several of them may play a role for the coupling to the chaotic sea, giving rise to multi-resonance transitions across subsequent resonance chains in phase space [27, 28]. As was argued in the context of near-integrable systems [28], such multi-resonance processes are indeed expected to dominate over couplings involving only one single resonance in the deep semiclassical limit ~ → 0.
The description of the coupling process across several consecutive resonances requires a generalization of Eq. (8.19) describing the modified eigenstate due to resonance-induced admixtures. We restrict ourselves, for this purpose, to including only the first-order matrix
elements hn + r| ˆHres(r:s)|ni for each resonance [i.e., only the matrix elements with k = 1 in
Eq. (8.36)]. For the sake of clarity, we furthermore consider the particular case of coupling processes that start in the lowest locally quantized eigenmode with node number n = 0 (the generalization to initial n 6= 0 being straightforward). The prescription we use is to consider that, although the approximation (8.15) is valid for only one resonance at a time, it is possible to sum the contributions obtained from different resonances. Considering a
sequence of consecutive r:s, r′:s′, r′′:s′′ . . . resonances, we obtain in this way
|ψ0i = |0i + X k>0 k Y l=1 hlr| ˆHres(r:s)|(l − 1)ri E0− Elr+ ls~ω ! × × ( |kri +X k′>0 k′ Y l′=1 hkr + l′r′| ˆH(r′ :s′ ) res |kr + (l′− 1)r′i E0− Ekr+l′r′+ (ks + l′s′)~ω ! × × " |kr + k′r′i + X k′′>0 k′′ Y l′′=1 hkr + k′r′+ l′′r′′| ˆH(r′′:s′′) res |kr + k′r′ + (l′′− 1)r′′i E0 − Ekr+k′r′+l′′r′′+ (ks + l′s′+ l′′s′′)~ω ! × × |kr + k′r′+ k′′r′′i + X k′′′>0 . . . !#) (8.37)
for the modified “ground state” within the island. Given an excited quasi-mode n far from
the interior of the island, the overlap hn|ψ0i obtained from Eq. (8.37) will in most cases
be exponentially dominated by one or a few contributions. There is no systematic way to identify them a priori, although some guiding principle can be used in this respect [28].
Quite naturally, for instance, low-order resonances, with comparatively small r and s, will, in general, give larger contribution than high-order resonances with comparable winding numbers s/r but larger r and s, due to the strong differences in the sizes of the mean
coupling matrix elements Vr:s [see, e.g., Eq. (8.22)]. In the same way, sequences of couplings
involving small denominators, i.e. energy differences like Eq. (8.21) that are close to zero, and thus intermediate steps symmetrically located on each side of a resonance, will tend to give larger contributions. In the small ~ limit this will tend to favor multi-resonance processes. Conversely, for intermediate values of ~ (in terms of the area of the regular region) the main contributions can be obtained from the lowest-order resonances. With few exceptions — especially concerning low-order resonances that are located close to the center of the island, thereby leading to relatively large energy denominators and small admixtures — this rule is generally observed for the semiclassical calculation of the eigenphase splittings we shall
consider in section 8.4.
8.3
The combination with chaos-assisted tunneling
We now discuss the implication of such nonlinear resonances on the tunneling process between the two symmetry-related regular islands under consideration. In the quantum system, these islands support (for not too large values of ~) locally quantized eigenstates or “quasimodes” with different node numbers n, which, due to the symmetry, have the same eigenvalues in both islands. In our case of a periodically driven system, these eigenvalues can be the
eigenphases ϕn of the unitary time evolution (Floquet) operator ˆU over one period τ of the
driving, or, alternatively, the quasienergies En such that ϕn= −Enτ /~ (modulo 2π).
The presence of a small (tunneling-induced) coupling between the islands lifts the de-generacy of the eigenvalues and yields the symmetric and antisymmetric linear combination of the quasimodes in the two islands as “true” eigenstates of the system. A nonvanishing
splitting ∆ϕn ≡ |ϕ+n− ϕ−n| consequently arises between the eigenphases ϕ±n of the symmetric
and the antisymmetric state, which is related to the splitting ∆En ≡ |En+ − En−| of the
quasi-energies E±
n through ∆ϕn = τ ∆En/~.
8.3.1 Resonance-assisted tunneling in near-integrable systems
We start by considering a system in the nearly integrable regime. In that case, we can
assume the presence of a (global) integrable Hamiltonian H0(p, q) that describes the dynamics
in the entire phase space to a very good approximation7. The energy splittings for the
corresponding quantum Hamiltonian ˆH0 ≡ H0(ˆp, ˆq) can be semiclassically calculated via an
analytic continuation of the invariant tori to the complex domain [8]. This generally yields the splittings
∆En(0) = ~Ωn
π exp(−σn/~) (8.38)
(up to a numerical factor of order one) where Ωn is the classical oscillation frequency
associ-ated with the nth quantized torus and σn denotes the imaginary part of the action integral
along the complex path that joins the two symmetry-related tori.
The main effect of nonlinear resonances in the non-integrable system is, as was discussed in the previous subsections, to induce perturbative couplings between quasimodes of different excitation within the regular islands. For the nearly integrable systems this can already lead
to a substantial enhancement of the splittings ∆Enas compared to Eq. (8.38) [27, 28]. As can
be derived within quantum perturbation theory, the presence of a prominent r:s resonance modifies the splitting of the local “ground state” in the island (i.e., the state with vanishing
7
Formally, this Hamiltonian is not identical with the unperturbed approximation H0(I) introduced in section 8.2.1 as the
definition of the latter is restricted to one well only. It is obvious, however, that H0(I) can be determined from H0(p, q), e.g.
node number n = 0) according to ∆ϕ0 = ∆ϕ(0)0 + kc X k=1 |A(r:s)kr |2∆ϕ (0) kr (8.39)
(using ∆ϕ(0)n ≫ ∆ϕ(0)0 for n > 0), where A
(r:s)
kr ≡ hkr|ψ0i denotes the admixture of the
(kr)th excited unperturbed component |kri to the perturbed ground state |ψ0i according to
Eq. (8.19) [possibly using Eq. (8.36) instead of (8.18)]. The maximal number kc of coupled
states is provided by the finite size of the island according to
kc =
1 r
area of the island
2π~ −
1 2
(8.40)
where the bracket stands for the integer part. The rapid decrease of the amplitudes A(r:s)kr
with k is compensated by an exponential increase of the unperturbed splittings ∆ϕ(0)kr, arising
from the fact that the tunnel action σn in Eq. (8.38) generally decreases with increasing n.
The maximal contribution to the modified ground state splitting is generally provided by the
state |kri for which Ikr+ I0 ≃ 2Ir:s— i.e., which in phase space is most closely located to the
torus that lies symmetrically on the opposite side of the resonance chain. This contribution is particularly enhanced by a small energy denominator [see Eq. (8.21)] and typically dominates the sum in Eq. (8.39).
As one goes further in the semiclassical ~ → 0 limit, a multi-resonance process is usually the dominant one. Neglecting interference terms between different coupling pathways that connect the ground state with a given excited mode |ni (which is justified due to the fact that the amplitudes associated with those coupling pathways are, in general, much different from each other in size), we obtain from Eq. (8.37) an expression of the form
∆ϕ0 = ∆ϕ(0)0 + X k |A(r:s)0,kr| 2∆ϕ(0) kr + X k X k′ |A(r:s)0,kr| 2 |A(rkr,kr+k′:s′) ′r′|2∆ϕ (0) kr+k′r′ + . . . (8.41)
with the coupling amplitudes
A(r:s)0,kr = k Y l=1 hlr| ˆHres(r:s)|(l − 1)ri E0− Elr+ ls~ω (8.42) A(rkr,kr+k′:s′) ′ r′ = k′ Y l′=1 hkr + l′r′| ˆH(r′:s′) res |kr + (l′− 1)r′i E0− Ekr+l′r′ + (ks + l′s′)~ω (8.43) A(rkr+k′′:s′′′r)′,kr+k′r′+k′′r′′ = . . .
for the eigenphase splitting.
8.3.2 Coupling with the chaotic sea
Turning now to the mixed regular-chaotic case, the integrable Hamiltonian H0(I) provides
invariant tori exist only up to a maximum action variable Ic corresponding to the outermost
boundary of the regular island in phase space. Beyond this outermost invariant torus, mul-tiple overlapping resonances provide various couplings and pathways such that unperturbed states in this regime can be assumed to be strongly connected to each other. Under such circumstances, it is natural to divide the Hilbert space into two parts, integrable and chaotic, associated respectively with the phase space regions within and outside the regular island.
For each symmetry class ± of the problem, let us introduce an effective Hamiltonian ˆHeff±
modeling the tunneling process. Let us furthermore denote ˆPregand ˆPch the (orthogonal)
pro-jectors onto the regular and chaotic Hilbert spaces. The diagonal blocks ˆH±
reg ≡ ˆPregHˆeff±Pˆreg
and Hch± ≡ ˆPchHˆeff±Pˆch receive a natural interpretation: within ˆHreg± , on the one hand, the
dynamics is exactly the same as in the nearly integrable regime above; ˆHch±, on the other
hand, is best modeled in a statistical manner by the introduction of random matrix ensem-bles. The only remaining delicate point is thus to connect the two, namely to model the
off-diagonal block ˆPregHˆeff±Pˆch. We stress that there is not yet a real consensus on the best
way how to do this, although various approaches give good quantitative accuracy.
To state more clearly the problem, let us consider a regular state |¯ni with quasi-energy
E0
¯
n close to the regular-chaos boundary. (Note that “close” here means that no resonance
within the island can connect |¯ni to a state |n′i within the island with n′ > ¯n. This notion of
“closeness” to the boundary is therefore ~-dependent.) The resonance assisted mechanism will connect any quasi-mode deep inside the island to such a state at the edge of the island. But to complete the description of the chaos-assisted tunneling process it is necessary to
compute the variance v2
¯
n of the random matrix elements vni¯ between |¯ni and the eigenstates
|ψc
ii of ˆHch± (the variance is independent of i if ˆHch± is modeled by the Gaussian orthogonal
or unitary ensemble).
One possible approach to compute this quantity is the fictitious integrable system ap-proach that was proposed by B¨acker et al. [22]. This method relies on the fact that, for the
effective Hamiltonian ˆHeff, the “direct” transition rate from a regular state |ni to the chaotic
region is given using Fermi’s golden rule by
Γdn→chaos = 2π ~
vn2
∆ch
, (8.44)
(see e.g. section 5.2.2 of Ref. [11] for a discussion in the context of random matrix theory)
where ∆ch denotes the mean spacing between eigenenergies within the chaotic block. As a
consequence, one obtains in first order in τ v2
n/~∆ch, || ˆPchU |ni||ˆ 2 = τ Γdn→chaos = 2πτ ~ v2n ∆ch (8.45)
(see also the contribution of B¨acker, Ketzmerick, and L¨ock in this book). If one can explicitly
construct a good integrable approximation Hreg ≡ H0(p, q) of the time-dependent dynamics
(see Sec. 8.2.1), this allows one, by quantum or semiclassical diagonalization, to determine
the unperturbed eigenstates |ni within the regular island, and to construct the projectors ˆPreg
and ˆPchThe “direct” regular-to-chaotic tunneling matrix elements of the nth quantized state
within the island is then evaluated by a simple application of the quantum time evolution
This approach can be qualified as “semi-numerical”, as it requires to numerically perform the quantum evolution for one period of the map (although this can be done by analyti-cal and semiclassianalyti-cal techniques in some cases [23], see also the contribution of B¨acker et
al. in this book). It strongly relies on the quality of the integrable approximation ˆHreg of
the Hamiltonian. If the latter was really diagonal (which, as a matter of principle, cannot be achieved by means of classical perturbation theory, due to the appearance of nonlinear resonances), Eq. (8.45) would represent an exact result (apart from the first-order
approxi-mation in τ v2
n/~∆ch which should not be a limitation in the tunneling regime). And indeed
very good agreement between this prediction and numerically computed tunneling rates was found for quantum maps that were designed such as to yield a “clean” mixed regular-chaotic phase space, containing a regular island and a chaotic region which both do not exhibit appreciable substructures [22], as well as for the mushroom billiard [23]. In more generic sit-uations, where nonlinear resonances are manifested within the regular island, this approach yields reliable predictions for the direct tunneling of regular states at the regular-chaos bor-der, and its combination with the resonance assisted mechanism described in the previous sections leads to good quantitative predictions for the tunneling rates for the states deep in the regular island [35]. We shall illustrate this on the example of the kicked rotor system in section 8.4.
8.3.3 Integrable semiclassical models for the regular-to-chaotic coupling
For now, however, we shall discuss other possible approaches of purely semiclassical nature (i.e. not involving any numerical evolution nor diagonalization of the quantum system)
to the calculation of the coupling parameter vn¯ for a regular state |¯ni at the edge of the
regular-chaos boundary.
One way to obtain an order of magnitude of v2
¯
nis to consider the decay of the quasimode
inside the regular island. For this purpose, let us assume that an integrable approximation
H0(I), valid up to a maximum action Ic corresponding to the chaos boundary, has been
ob-tained. Within the Birkhoff-Gustavson normal-form coordinates (P, Q) given by Eqs. (8.28)
and (8.29), H0 appears as a function of the harmonic-oscillator Hamiltonian (P2+ Q2)/2
and therefore has the same eigenstates
hQ| ¯ni = √1 2¯nn!¯ 1 (π~)1/4 exp −Q 2/2~ H ¯ n(Q/~1/2) (8.46) ≃ 1 2p2π|Pn¯(Q)| exp − Z Q Q1 |Pn¯(Q)|dQ for Q > Qn , (8.47)
where H¯n are the Hermite polynomials and P¯n(Q) = p2I¯n− Q2. The last equation
cor-responds to the semiclassical asymptotics in the forbidden region on the right-hand side
of the turning point Q1 = √2I¯n with I¯n = ~(¯n + 1/2). In Q representation, the regular
island extends up to Qc = √2Ic, at which point de state |¯ni has decayed to ψn¯(Qc) ≃
1 2√2π|Pn¯(Qc)| exp (−S(I¯n , Ic)), where S(I¯n, Ic) = Z Qc Q1 |Pn¯(Q)|dQ =pIc(Ic− In¯) − Inln (pIc − In¯ +pIc)/pI¯n . (8.48)
This expression is suspiciously simple (as it depends only on I¯n and Ic, and on no other
property of the system), and should not be taken too seriously as it is.
Indeed, it should be borne in mind that v2
¯
n is related not so much to the value of the
wavefunction at the regular-chaos boundary than to the transition rate Γd
n→chaos through
Eq. (8.44), which we can equal to the current flux Jn¯ through this boundary for the regular
state |¯ni. Using H0(I) to compute this current leads to a zero result and it is therefore
mandatory to use a better approximation of the non-integrable Hamiltonian H to obtain a meaningful answer. What complicates the evaluation of the transition rate from the edge of the regular to the chaotic domain is therefore that one needs to find an approximation describing both the regular and chaotic dynamics — unless one actually uses there the exact quantum dynamics as was done in Ref. [22]. Since the regular-chaos border is typically the place where approximation schemes tend to be difficult to control, this will rely on some assumption to be made for the chaotic regions, two possible choices of which we shall describe now.
One scenario that has been considered amounts in some way to model the regular to chaos
transition in the way depicted in Fig. 8.4a: a kinetic-plus-potential Hamiltonian p2/2m+V (q)
where the island itself correspond to a potential well and the edge of the regular region to the place where the potential decreases abruptly. The picture one has in mind in that case is that escaping from the edge of the regular island to the chaotic sea is akin to the standard textbook barrier tunneling [45]. Using Langer’s connection formula [46] within that model,
the semiclassical wavefunction for the quasi-mode ¯n inside the potential well can be extended
under the potential barrier and, beyond this, into the region where motion at energy E0
¯ n is
again classically authorized. In the classically allowed region outside the well (q > q′
r) the
semiclassical wavefunction can be written as
ψn¯(q) ≃ s Ω¯n 2πp¯n(q) exp i ~ Z q q′ r p¯n(q)dq + iπ/4 exp −S~t , (8.49)
with Ωn¯ the angular frequency of the torus E¯n0, p¯n(q) =p2m[En0¯− V (q)], and
St =
Z q′r
qr
|p¯n(q)|dq (8.50)
the tunneling action8. The current of probability leaving the well is then given by
Jn¯ ≡ ~ m Im[ψ ∗ ¯ n∇ψ¯n] = Ωn¯ 2π exp −2S~t , (8.51) from which v2 ¯
n is obtained through Eq. (8.44) (identifying J¯n with Γdn→chaos¯ ).
Although Eq. (8.51) is derived here for the particular case of a kinetic-plus-potential Hamiltonian, it applies more generally (up maybe to factors of order one) to any system with a phase space portrait that is similar to the one of Fig. 8.4b, where tori inside the island can be analytically continued in the complex plane to a manifold escaping to infinity.
8
Strictly speaking, outgoing (Siegert) boundary conditions [47] need to be employed in Eq. (8.49) in order to properly describe the decay process from the well. Those outgoing boundary conditions involve, in addition, an exponential increase of the wavefunction’s amplitude with increasing distance from the well, which is not taken into account in Eq. (8.49) assuming that the tunneling rate from the well is comparatively weak.
q V (q) q p En0 qr q′ r qM
Figure 8.4: Modeling of the direct coupling between the edge of the regular region to the chaotic one by a potential barrier separating a potential well from an “open” region. (a) Sketch of the potential. (b) Corresponding phase space portrait.
In that case, Eq. (8.51) can be applied provided the tunneling action St is taken as the
imaginary part of the action integral on a path joining the interior to the exterior of the island on this analytical continuation. As a last approximation, one may assume that the transition to the “open” part is extremely sharp once the separatrix is crossed. In the model of Fig. 8.4a, this amounts to assume a very rapid decrease of the potential, in which case
one may replace qr by qM in the tunneling action Eq. (8.50). In an actual calculation of the
direct tunneling rate for a regular state at the edge of the chaos boundary (which we can reliably describe only coming from the interior of the island) this amounts to consider in the
same way that the action S(I¯n, Ic) [Eq. (8.48)] provides a good approximation to St. Under
this hypothesis, one obtains for the coupling to the chaotic Hilbert space the prediction
vn2¯ = ∆ch~Ωn¯ 4π2 exp −2S(In¯~, Ic) (8.52)
(see [48] where this computation was proposed with the slightly different language of complex time trajectories).
This “potential-barrier” picture of the direct tunneling is in essence what is behind the approach of Podolskiy and Narimanov [19] (though their treatment of the problem is a bit
more sophisticated). Its main virtue is that, beyond the quantized action I¯n = ~(¯n + 1/2),
Eq. (8.52) relies only on simple characteristics of the integrable regions : its area 2πIcand the
assumes that the direct tunneling mechanism corresponds in some sense to the phase portrait of Fig. 8.4b, which in general cannot be justified within any controlled approximation scheme.
8.3.4 Regular-to-chaotic coupling via a nonlinear resonance
Another possible, and presumably more realistic, approach to evaluate the direct coupling
parameter vn¯ can be obtained assuming that the effective model (8.15) describes the vicinity
of a resonance not only inside the regular region, but also within the chaotic sea in the near vicinity of the island. In this perspective, the model one has in mind for the chaotic region follows the spirit of Chirikov’s overlapping criterion for the transition to chaos [49, 39]: resonances still provide the couplings between quasi-modes, but in the chaotic region these couplings become strong enough to completely mix the states. Near the regular-chaos edge, the transition between modes inside and outside the regular region is still dominated by one or several r:s resonances which might be within or possibly outside the regular island.
As an illustration, let us consider the simple case where a single nonlinear r:s resonance is responsible for all couplings, both within the island and from the island’s edge to the chaotic region. Keeping in mind the discussion in Section 8.2.2 and in Ref. [28], we assume here that the couplings induced by the r:s resonance are dominantly described by the lowest
non-vanishing Fourier component V1 of the perturbation, i.e. by the matrix elements Vr:s(n+r) ≡
hn + r| ˆHeff|ni, and set the phase φ1 to zero without loss of generality.
The structure of the effective Hamiltonian that describes the coupling of the ground state
E0 to the chaotic sea is, in that case, given by
Heff± = ˜ E0 Vr:s(r) Vr:s(r) . .. . .. . .. ˜Ekcr V [(kc+1)r] r:s V[(kc+1)r] r:s chaos± , (8.53)
where ˜Ekr ≡ Ekr− Ωr:sIkr are the eigenenergies of the unperturbed Hamiltonian H0 in the
co-rotating frame, and the chaotic part (the square in the lower right corner) consists of a full sub-block with equally strong couplings between all basis states with actions beyond the outermost invariant torus of the islands. In this example the last state within the island
connected to the ground state is the quasi-mode ¯n = kcr and vn2¯ = (V
[(kc+1)r]
r:s )2/Nch, with
Nch the number of states with a given parity in the chaotic Hilbert space. In a more general
situation, many resonances may, as in Eq. (8.37), be involved in connecting the ground stated
to some |¯ni at the edge of the island, but we would still write the variance v2
¯
n of the matrix
elements providing the last coupling to the chaotic region as the ratio (V[(kc+1)r]
r:s )2/Nch for
We stress here that it is necessary in this approach to have an explicit access to the number
of states Nch in the chaotic Hilbert space. This is obtained quite trivially for quantum maps,
such as the kicked rotor we shall consider in section 8.4, when ~ = 2π/N with integer N.
In that case N is the total number of states in the full Hilbert space and Nch/N represents
the relative area of the chaotic region in phase space. For a two dimensional conservative
Hamiltonian, on the other hand, N and thus Nch are related to the Thouless energy (see
e.g. the section 2.1.4 of [50]), but this provides only an energy scale rather than a precise number, and a more detailed discussion is required to be quantitative. For the sake of clarity, we shall in the following limit our discussion to the simpler case that N is known.
8.3.5 Theory of chaos-assisted tunneling
Let us consider now the effect of the chaotic block on the tunneling process. Eliminating intermediate states within the regular island leads for the effective Hamiltonian a matrix of the form Heff± = E0 Veff 0 · · · 0 Veff H11± · · · H1N±ch 0 ... ... .. . ... ... 0 HN±ch1 · · · HN±chNch . (8.54)
for each symmetry class. In the simplest case Eq. (8.53) where a single r:s resonance needs to be considered, the effective coupling matrix element between the ground state and the
chaos block (Hij±) is given by
Veff = Vr:s[(kc+1)r] kc Y k=1 Vr:s(kr) E0− Ekr+ ks~ω (8.55)
where Enare the unperturbed energies (8.20) of Heff and |kcri represents the highest
unper-turbed state that is connected by the r:s resonance to the ground state and located within
the island (i.e., Ikcr < Ic < I(kc+1)r). More generally, Veff can be expressed in terms of the
couplings associated with the various resonances that contribute to the transitions within the island and at the regular-chaos edge. The form of this expression (8.55) already provides an explanation for the appearance of plateau-like structures in the tunneling rates. Indeed,
decreasing ~ leads to discontinous increments of the maximal number kcof couplings through
Eq. (8.40) and hence to step-like reductions of the effective matrix element Veff, while in
be-tween such steps Veff varies smoothly through the action dependence of the coupling matrix
elements Vr:s(kr), provided accidential near-degeneracies in the energy denominators do not
occur.
In the simplest possible approximation, which follows the lines of Refs. [12, 17], we neglect the effect of partial barriers in the chaotic part of the phase space [11] and assume that the
chaos block (Hij±) is adequately modeled by a random Hermitian matrix from the Gaussian
orthogonal ensemble (GOE). After a pre-diagonalization of (Hij±), yielding the eigenstates
antisymmetric ground state energies by E0± = E0+ Nch X j=1 |vjeff±|2 E0− Ej± , (8.56)
with veff±j ≡ Veffhkr|φ±ji Performing the random matrix average for the eigenvectors, we
obtain that hh|hkr|φ±j i|2ii ≃ 1/Nch for all j = 1 . . . Nch, which simply expresses the fact that
none of the basis states is distinguished within the chaotic block (Hij±). As a consequence,
the variance of the veff±j ’s is independent of j and equal to v2
eff = Veff2/Nch.
As was shown in Ref. [17], the random matrix average over the eigenvalues Ej± gives rise
to a Cauchy distribution for the shifts of the ground state energies, and consequently also for the splittings
∆E0 = |E0+− E0−| (8.57)
between the symmetric and the antisymmetric ground state energy. For the latter, we specifically obtain the probability distribution
P (∆E0) = 2 π ∆E0 (∆E0)2 + (∆E0)2 (8.58) with ∆E0 = 2πv2 eff ∆ch (8.59)
where ∆ch denotes the mean level spacing in the chaos at energy E0. This distribution is,
strictly speaking, valid only for ∆E0 ≪ veff and exhibits a cutoff at ∆E0 ∼ 2veff, which
ensures that the statistical expectation value h∆E0i =
R∞
0 xP (x)dx does not diverge.
Since tunneling rates and their parametric variations are typically studied on a logarithmic
scale [i.e., log(∆E0) rather than ∆E0is plotted vs. 1/~], the relevant quantity to be calculated
from Eq. (8.58) and compared to quantum data is not the mean value h∆E0i, but rather the
average of the logarithm of ∆E0. We therefore define our “average” level splitting h∆E0ig
as the geometric mean of ∆E0, i.e.
h∆E0ig ≡ exp [hln(∆E0)i] (8.60)
and obtain as result the scale defined in Eq. (8.59),
h∆E0ig = ∆E0. (8.61)
This expression further simplifies for our specific case of periodically driven systems,
where the time evolution operator ˆU is modeled by the dynamics under the effective
Hamil-tonian (8.54) over one period τ . In this case, the chaotic eigenphases Ej±τ /~ are uniformly
distributed in the interval [0, 2π[. We therefore obtain
∆ch =
2π~
Nchτ
for the mean level spacing near E0. This yields h∆ϕ0ig ≡ τ ~ h∆E0ig = τ Veff ~ 2 (8.63)
for the geometric mean of the ground state’s eigenphase splitting. Note that this final result
does not depend on the number Nch of chaotic states within the sub-block (Hij±); as long as
this number is sufficiently large to justify the validity of the Cauchy distribution (8.58) (see Ref. [17]), the geometric mean of the eigenphase splitting is essentially given by the square
of the effective coupling Veff from the ground state to the chaos.
The distribution (8.58) also permits the calculation of the logarithmic variance of the eigenphase splitting: we obtain
[ln(∆ϕ0) − hln(∆ϕ0)i]2 =
π2
4 . (8.64)
This universal result predicts that the actual splittings may be enhanced or reduced compared
to h∆ϕ0ig by factors of the order of exp(π/2) ≃ 4.8, independently of the values of ~ and
external parameters. Indeed, as was discussed in Ref. [32], short-range fluctuations of the splittings, arising at small variations of ~, are well characterized by the standard deviation that is associated with Eq. (8.64).
It is interesting to note that the expression (8.63) for the (geometric) mean level spacing is quantitatively identical with the expression (8.45) for the mean escape rate from the regular island to the chaotic sea derived in Ref. [22] using Fermi’s golden rule. This seems surprising as two different nonclassical processes, namely Rabi oscillations between equivalent islands and the decay from an island within an open system, underly these expressions. In one-dimensional single-barrier tunneling problems, these two processes would indeed give rise to substantially different rates; in Eq. (8.1), to be more precise, the imaginary action integral in the exponent would have to be multiplied by two in order to obtain the corresponding expres-sion for the decay rate (and the overall prefactor in front of the exponential function should be divided by two, which is not important here). The situation is a bit different, however, in our case of dynamical tunneling in mixed regular-chaotic systems. In such systems, level splittings between two equivalent regular islands involve two identical dynamical tunneling processes between the islands and the chaotic sea (namely one process for each island), while the decay into the chaotic sea involves only one such process, with, however, the square of the corresponding (exponentially suppressed) coupling coefficient. This explains from our point of view the equivalence of the expressions (8.63) and (8.45).
We finally remark that the generalization of the expression for the mean splittings to multi-resonance processes is straightforward and amounts to replacing the product of admixtures in Eq. (8.55) by a product involving several resonances subsequently, in close analogy with Eq. (8.41). In fact, the multi-resonance expression (8.41) can be directly used in this context
replacing the “direct” splittings ∆ϕ(0)n by (Vr[(kf:scf+1)rf]τ /~)
2 where the r
f:sf resonance is the
one that induces the final coupling step to the chaotic sea (provided In < Ic < In+rf holds
for the corresponding action variables; otherwise we would set ∆ϕ(0)n = 0). This expression
represents the basic formula that is used in the semiclassical calculations of the splittings in the kicked rotor model, to be discussed below.
8.3.6 The role of partial barriers in the chaotic domain
In the previous section, we assumed a perfectly homogeneous structure of the Hamiltonian outside the outermost invariant torus, which allowed us to make a simple random-matrix ansatz for the chaotic block. This assumption hardly ever corresponds to reality. As was shown in Refs. [51, 11, 12] for the quartic oscillator, the chaotic part of the phase space is, in general, divided into several subregions which are weakly connected to each other through partial transport barriers for the classical flux (see, e.g., Fig. 8 in Ref. [12]) This substructure of the chaotic phase space (which is generally not visible in a Poincar´e surface of section) is particularly pronounced in the immediate vicinity of a regular island, where a dense hierarchical sequence of partial barriers formed by broken invariant tori and island chains is accumulating [52, 53, 54].
In the corresponding quantum system, such partial barriers may play the role of “true” tunneling barriers in the same spirit as invariant classical tori. This will be the case if the phase space area ∆W that is exchanged across such a partial barrier within one classical iteration is much smaller than Planck’s constant 2π~ [55], while in the opposite limit ∆W ≫
2π~ the classical partial barrier appears completely transparent in the quantum system9.
Consequently, the “sticky” hierarchical region around a regular island acts, for not extremely small values of ~, as a dynamical tunneling area and thereby extends the effective “quantum” size of the island in phase space. As a matter of fact, this leads to the formation of localized states (also called “beach” states in the literature [13]) which are supported by this sticky phase space region in the surrounding of the regular island [56] (see Fig. 8.6b).
An immediate consequence of the presence of such partial barriers for resonance-assisted
tunneling is the fact that the critical action variable Ic defining the number kc of
resonance-assisted steps within the island according to Eq. (8.55) should not be determined from the outermost invariant torus of the island, but rather from the outermost partial barrier that acts like an invariant torus in the quantum system. We find that this outermost quantum barrier is, for not extremely small values of ~, generally formed by the stable and unstable manifolds that emerge from the hyperbolic periodic points associated with a low-order non-linear r:s resonance. These manifolds are constructed until their first intersection points in between two adjacent periodic points, and iterated r − 1 times (or r/2 − 1 times in the case of period-doubling of the island chain due to discrete symmetries), such as to form a closed
artificial boundary around the island in phase space10 As shown in Fig. 8.5, one further
iteration maps this boundary onto itself, except for a small piece that develops a loop-like deformation. The phase space area enclosed between the original and the iterated boundary precisely defines the classical flux ∆W exchanged across this boundary within one iteration of the map [52, 53].
The example in Fig. 8.5 shows a boundary that arises from the inner stable and unstable manifolds (i.e. the ones that would, in a near-integrable system, form the inner separatrix structure) emerging from the unstable periodic points of a 4:1 resonance (which otherwise is not visible in the Poincar´e section) in the kicked rotor system. Judging from the size of the flux area ∆W , this boundary should represent the relevant quantum chaos border for the tunneling processes that are discussed in the following section. We clearly see that it
9
More precisely, the authors of Ref. [55] claim that ∆W has to be compared with π~ in order to find out whether or not a given partial barrier is transparent in the quantum system.
10
This construction is also made in order to obtain the phase space areas S±r:s that are enclosed by the outer and inner
separatrix structures of an r:s resonance, and that are needed in order to compute the mean action variable Ir:s and the
